reserve x,y for Real;
reserve a,b,c for Element of Real_Lattice;
reserve p,q,r for Element of Real_Lattice;
reserve A,B for non empty set;
reserve f,g,h for Element of Funcs(A,REAL);

theorem Th10:
  (maxfuncreal(A)).((maxfuncreal(A)).(f,g),h) =(maxfuncreal(A)).(f
  ,(maxfuncreal(A)).(g,h))
proof
  now
    let x be Element of A;
A1: x in dom (maxreal.:(f,g)) by Lm6;
A2: x in dom (maxreal.:(g,h)) by Lm6;
A3: x in dom (maxreal.:((maxreal.:(f,g)),h)) by Lm6;
A4: x in dom (maxreal.:(f,(maxreal.:(g,h)))) by Lm6;
    thus ((maxfuncreal(A)).((maxfuncreal(A)).(f,g),h)).x =((maxfuncreal(A)).(
    maxreal.:(f,g),h)).x by Def4
      .=(maxreal.:(maxreal.:(f,g),h)).x by Def4
      .=maxreal.((maxreal.:(f,g)).x,h.x) by A3,FUNCOP_1:22
      .=maxreal.(maxreal.(f.x,g.x),h.x) by A1,FUNCOP_1:22
      .=maxreal.(f.x,maxreal.(g.x,h.x)) by Th3
      .=maxreal.(f.x,((maxreal.:(g,h)).x)) by A2,FUNCOP_1:22
      .=(maxreal.:(f,(maxreal.:(g,h)))).x by A4,FUNCOP_1:22
      .=((maxfuncreal(A)).(f,(maxreal.:(g,h)))).x by Def4
      .=((maxfuncreal(A)).(f,((maxfuncreal(A)).(g,h)))).x by Def4;
  end;
  hence thesis by FUNCT_2:63;
end;
